How does the small-scale topological structure of an airline network behave as the network evolves? To address this question, we study the dynamic properties of small undirected subgraphs using 15 years of data on Southwest Airlines' domestic route service. We use exact enumeration formulae to identify statistically over- and under-represented subgraphs, known as motifs and anti-motifs. We discover substantial topology transitions in Southwest's network and provide evidence for time-varying power-law scaling between subgraph counts and the number of edges in the network. We also suggest a node-ranking measure that can identify important nodes relative to specific local topologies. Our results extend the toolkit of subgraph-based methods and provide new insight into transportation networks and the strategic behaviour of firms.
Many real world networks contain a statistically surprising number of certain subgraphs, called network motifs. In the prevalent approach to motif analysis, network motifs are detected by comparing subgraph frequencies in the original network with a statistical null model. In this paper we propose an alternative approach to motif analysis where network motifs are defined to be connectivity patterns that occur in a subgraph cover that represents the network using minimal total information. A subgraph cover is defined to be a set of subgraphs such that every edge of the graph is contained in at …
Dense regions in networks are an indicator of interesting and unusual information. However, most existing methods only consider simple, undirected, unweighted networks. Complex networks in the real-world often have rich information though: edges are asymmetrical and nodes/edges have categorical and numerical attributes. Finding dense subgraphs in such networks in accordance with this rich information is an important problem with many applications. Furthermore, most existing algorithms ignore the higher-order relationships (i.e., motifs) among the nodes. Motifs are shown to be helpful for dense…
Exploring statistics of locally connected subgraph patterns (also known as network motifs) has helped researchers better understand the structure and function of biological and online social networks (OSNs). Nowadays the massive size of some critical networks -- often stored in already overloaded relational databases -- effectively limits the rate at which nodes and edges can be explored, making it a challenge to accurately discover subgraph statistics. In this work, we propose sampling methods to accurately estimate subgraph statistics from as few queried nodes as possible. We present samplin…
The observation that some subgraphs, called motifs, appear more often in real networks than in their randomized counterparts has attracted much attention in the scientific community. In the prevalent approach the detection of motifs is based on comparing subgraph counts in a network with their counterparts in the configuration model with the same degree distribution as the network. In this short note we derive conservation laws that relate motif counts in the configuration model.
Motivated by applications in social network community analysis, we introduce a new clustering paradigm termed motif clustering. Unlike classical clustering, motif clustering aims to minimize the number of clustering errors associated with both edges and certain higher order graph structures (motifs) that represent "atomic units" of social organizations. Our contributions are two-fold: We first introduce motif correlation clustering, in which the goal is to agnostically partition the vertices of a weighted complete graph so that certain predetermined "important" social subgraphs mostly lie with…
Tenuous subgraph finding aims to detect a subgraph with few social interactions and weak relationships among nodes. Despite significant efforts have been made on this task, they are mostly carried out in view of graph-structured data. These methods depend on calculating the shortest path and need to enumerate all the paths between nodes, which suffer the combinatorial explosion. Moreover, they all lack the integration of neighborhood information. To this end, we propose a novel model named Graph Neural Network with Motif-aware for tenuous subgraph finding (GNNM), a neighborhood aggregation bas…
For scale-free networks with degrees following a power law with an exponent $τ\in(2,3)$, the structures of motifs (small subgraphs) are not yet well understood. We introduce a method designed to identify the dominant structure of any given motif as the solution of an optimization problem. The unique optimizer describes the degrees of the vertices that together span the most likely motif, resulting in explicit asymptotic formulas for the motif count and its fluctuations. We then classify all motifs into two categories: motifs with small and large fluctuations.
The structure of complex networks can be characterized by counting and analyzing network motifs. Motifs are small subgraphs that occur repeatedly in a network, such as triangles or chains. Recent work has generalized motifs to temporal and dynamic network data. However, existing techniques do not generalize to sequential or trajectory data, which represents entities moving through the nodes of a network, such as passengers moving through transportation networks. The unit of observation in these data is fundamentally different, since we analyze full observations of trajectories (e.g., a trip fr…
A growing set of applications consider the process of network formation by using subgraphs as a tool for generating the network topology. One of the pressing research challenges is thus to be able to use these subgraphs to understand the network topology of information cascades which ultimately paves the way to theorize about how information spreads over time. In this paper, we make the first attempt at using network motifs to understand whether or not they can be used as generative elements for the diffusion network organization during different phases of the cascade lifecycle. In doing so, w…
Algorithms for mining very large graphs, such as those representing online social networks, to discover the relative frequency of small subgraphs within them are of high interest to sociologists, computer scientists and marketeers alike. However, the computation of these network motif statistics via naive enumeration is infeasible for either its prohibitive computational costs or access restrictions on the full graph data. Methods to estimate the motif statistics based on random walks by sampling only a small fraction of the subgraphs in the large graph address both of these challenges. In thi…
Networks are a fundamental tool for modeling complex systems in a variety of domains including social and communication networks as well as biology and neuroscience. Small subgraph patterns in networks, called network motifs, are crucial to understanding the structure and function of these systems. However, the role of network motifs in temporal networks, which contain many timestamped links between the nodes, is not yet well understood. Here we develop a notion of a temporal network motif as an elementary unit of temporal networks and provide a general methodology for counting such motifs. We…
A $k$-motif (or graphlet) is a subgraph on $k$ nodes in a graph or network. Counting of motifs in complex networks has been a well-studied problem in network analysis of various real-word graphs arising from the study of social networks and bioinformatics. In particular, the triangle counting problem has received much attention due to its significance in understanding the behavior of social networks. Similarly, subgraphs with more than 3 nodes have received much attention recently. While there have been successful methods developed on this problem, most of the existing algorithms are not scala…
Investigating the frequency and distribution of small subgraphs with a few nodes/edges, i.e., motifs, is an effective analysis method for static networks. Motif-driven analysis is also useful for temporal networks where the spectrum of motifs is significantly larger due to the additional temporal information on edges. This variety makes it challenging to design a temporal motif model that can consider all aspects of temporality. In the literature, previous works have introduced various models that handle different characteristics. In this work, we compare the existing temporal motif models and…
Graphs are naturally used to describe the structures of various real-world systems in biology, society, computer science etc., where subgraphs or motifs as basic blocks play an important role in function expression and information processing. However, existing research focuses on the basic statistics of certain motifs, largely ignoring the connection patterns among them. Recently, a subgraph network (SGN) model is proposed to study the potential structure among motifs, and it was found that the integration of SGN can enhance a series of graph classification methods. However, SGN model lacks di…
Motifs are the fundamental components of complex systems. The topological structure of networks representing complex systems and the frequency and distribution of motifs in these networks are intertwined. The complexities associated with graph and subgraph isomorphism problems, as the core of frequent subgraph mining, have direct impacts on the performance of motif discovery algorithms. To cope with these complexities, researchers have adopted different strategies for candidate generation and enumeration, and frequency computation. In the past few years, there has been an increasing interest i…
Institute for Theoretical Physics, University of Würzburg（维尔茨堡大学理论物理研究所）
The detection of triadic subgraph motifs is a common methodology in complex-networks research. The procedure usually applied in order to detect motifs evaluates whether a certain subgraph pattern is overrepresented in a network as a whole. However, motifs do not necessarily appear frequently in every region of a graph. For this reason, we recently introduced the framework of Node-Specific Pattern Mining (NoSPaM). This work is a manual for an implementation of NoSPaM which can be downloaded from www.mwinkler.eu.
Is it possible to have a unique, recognizable style in soccer nowadays? We address this question by proposing a method to quantify the motif characteristics of soccer teams based on their pass networks. We introduce the the concept of "flow motifs" to characterize the statistically significant pass sequence patterns. It extends the idea of the network motifs, highly significant subgraphs that usually consists of three or four nodes. The analysis of the motifs in the pass networks allows us to compare and differentiate the styles of different teams. Although most teams tend to apply homogenous …
Characterizing motif (i.e., locally connected subgraph patterns) statistics is important for understanding complex networks such as online social networks and communication networks. Previous work made the strong assumption that the graph topology of interest is known, and that the dataset either fits into main memory or stored on disks such that it is not expensive to obtain all neighbors of any given node. In practice, researchers have to deal with the situation where the graph topology is unknown, either because the graph is dynamic, or because it is expensive to collect and store all topol…
Many real-world applications give rise to large heterogeneous networks where nodes and edges can be of any arbitrary type (e.g., user, web page, location). Special cases of such heterogeneous graphs include homogeneous graphs, bipartite, k-partite, signed, labeled graphs, among many others. In this work, we generalize the notion of network motifs to heterogeneous networks. In particular, small induced typed subgraphs called typed graphlets (heterogeneous network motifs) are introduced and shown to be the fundamental building blocks of complex heterogeneous networks. Typed graphlets are a power…
Federation University Australia（澳大利亚联邦大学）；Dalian University of Technology（大连理工大学）；Swinburne University of Technology（斯威本科技大学）；Deakin University（迪肯大学）；University of South Australia（南澳大利亚大学）
Clustering a group of vertices in networks facilitates applications across different domains, such as social computing and Internet of Things. However, challenges arises for clustering networks with increased scale. This paper proposes a solution which consists of two motif clustering techniques: standard acceleration CHIEF-ST and approximate acceleration CHIEF-AP. Both algorithms first find the maximal k-edge-connected subgraphs within the target networks to lower the network scale, then employ higher-order motifs in clustering. In the first procedure, we propose to lower the network scale by…
Massively parallel architectures such as the GPU are becoming increasingly important due to the recent proliferation of data. In this paper, we propose a key class of hybrid parallel graphlet algorithms that leverages multiple CPUs and GPUs simultaneously for computing k-vertex induced subgraph statistics (called graphlets). In addition to the hybrid multi-core CPU-GPU framework, we also investigate single GPU methods (using multiple cores) and multi-GPU methods that leverage all available GPUs simultaneously for computing induced subgraph statistics. Both methods leverage GPU devices only, wh…
The statistical significance of network properties is conditioned on null models which satisfy spec- ified properties but that are otherwise random. Exponential random graph models are a principled theoretical framework to generate such constrained ensembles, but which often fail in practice, either due to model inconsistency, or due to the impossibility to sample networks from them. These problems affect the important case of networks with prescribed clustering coefficient or number of small connected subgraphs (motifs). In this paper we use the Wang-Landau method to obtain a multicanonical s…
China University of Geosciences (Beijing)（中国地质大学（北京））；University of Naples Federico II（那不勒斯费德里科二世大学）
Clustering analysis has been widely used in trust evaluation on various complex networks such as wireless sensors networks and online social networks. Spectral clustering is one of the most commonly used algorithms for graph-structured data (networks). However, the conventional spectral clustering is inherently difficult to work with large-scale networks due to the fact that it needs computationally expensive matrix manipulations. To deal with large networks, in this paper, we propose an efficient graph clustering algorithm, KCoreMotif, specifically for large networks by exploiting k-core deco…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Auburn University（奥本大学）
Motivated by applications in social and biological network analysis, we introduce a new form of agnostic clustering termed~\emph{motif correlation clustering}, which aims to minimize the cost of clustering errors associated with both edges and higher-order network structures. The problem may be succinctly described as follows: Given a complete graph $G$, partition the vertices of the graph so that certain predetermined `important' subgraphs mostly lie within the same cluster, while `less relevant' subgraphs are allowed to lie across clusters. Our contributions are as follows: We first introduc…
Following the work of Krumov et al. [Eur. Phys. J. B 84, 535 (2011)] we revisit the question whether the usage of large citation datasets allows for the quantitative assessment of social (by means of coauthorship of publications) influence on the progression of science. Applying a more comprehensive and well-curated dataset containing the publications in the journals of the American Physical Society during the whole 20th century we find that the measure chosen in the original study, a score based on small induced subgraphs, has to be used with caution, since the obtained results are highly sen…
PageRank has been widely used to measure the authority or the influence of a user in social networks. However, conventional PageRank only makes use of edge-based relations, which represent first-order relations between two connected nodes. It ignores higher-order relations that may exist between nodes. In this paper, we propose a novel framework, motif-based PageRank (MPR), to incorporate higher-order relations into the conventional PageRank computation. Motifs are subgraphs consisting of a small number of nodes. We use motifs to capture higher-order relations between nodes in a network and in…
Many real-world phenomena are best represented as interaction networks with dynamic structures (e.g., transaction networks, social networks, traffic networks). Interaction networks capture flow of data which is transferred between their vertices along a timeline. Analyzing such networks is crucial toward comprehend- ing processes in them. A typical analysis task is the finding of motifs, which are small subgraph patterns that repeat themselves in the network. In this paper, we introduce network flow motifs, a novel type of motifs that model significant flow transfer among a set of vertices wit…
Network community detection is a hot research topic in network analysis. Although many methods have been proposed for community detection, most of them only take into consideration the lower-order structure of the network at the level of individual nodes and edges. Thus, they fail to capture the higher-order characteristics at the level of small dense subgraph patterns, e.g., motifs. Recently, some higher-order methods have been developed but they typically focus on the motif-based hypergraph which is assumed to be a connected graph. However, such assumption cannot be ensured in some real-worl…
Finding dense bipartite subgraphs and detecting the relations among them is an important problem for affiliation networks that arise in a range of domains, such as social network analysis, word-document clustering, the science of science, internet advertising, and bioinformatics. However, most dense subgraph discovery algorithms are designed for classic, unipartite graphs. Subsequently, studies on affiliation networks are conducted on the co-occurrence graphs (e.g., co-author and co-purchase) that project the bipartite structure to a unipartite structure by connecting two entities if they shar…
Pattern counting in graphs is fundamental to network science tasks, and there are many scalable methods for approximating counts of small patterns, often called motifs, in large graphs. However, modern graph datasets now contain richer structure, and incorporating temporal information in particular has become a critical part of network analysis. Temporal motifs, which are generalizations of small subgraph patterns that incorporate temporal ordering on edges, are an emerging part of the network analysis toolbox. However, there are no algorithms for fast estimation of temporal motifs counts; mor…
The mining of graphs in terms of their local substructure is a well-established methodology to analyze networks. It was hypothesized that motifs - subgraph patterns which appear significantly more often than expected at random - play a key role for the ability of a system to perform its task. Yet the framework commonly used for motif-detection averages over the local environments of all nodes. Therefore, it remains unclear whether motifs are overrepresented in the whole system or only in certain regions. In this contribution, we overcome this limitation by mining node-specific triad patterns. …
Wikipedia serves as a good example of how editors collaborate to form and maintain an article. The relationship between editors, derived from their sequence of editing activity, results in a directed network structure called the revision network, that potentially holds valuable insights into editing activity. In this paper we create revision networks to assess differences between controversial and non-controversial articles, as labelled by Wikipedia. Originating from complex networks, we apply motif analysis, which determines the under or over-representation of induced sub-structures, in this …
The lifestyles of urban dwellers could reveal important insights regarding the dynamics and complexity of cities. Despite growing research on analysis of lifestyle patterns in cities, little is known about the characteristics of people's lifestyles patterns at urban scale. This limitation is primarily due to challenges in characterizing lifestyle patterns when human movement data is aggregated to protect the privacy of users. In this study, we model cities based on aggregated human visitation data to construct a network of places. We then examine the subgraph signatures in the networks of plac…
Counting the number of occurrences of small connected subgraphs, called temporal motifs, has become a fundamental primitive for the analysis of temporal networks, whose edges are annotated with the time of the event they represent. One of the main complications in studying temporal motifs is the large number of motifs that can be built even with a limited number of vertices or edges. As a consequence, since in many applications motifs are employed for exploratory analyses, the user needs to iteratively select and analyze several motifs that represent different aspects of the network, resulting…
Conventionally, pairwise relationships between nodes are considered to be the fundamental building blocks of complex networks. However, over the last decade the overabundance of certain sub-network patterns, so called motifs, has attracted high attention. It has been hypothesized, these motifs, instead of links, serve as the building blocks of network structures. Although the relation between a network's topology and the general properties of the system, such as its function, its robustness against perturbations, or its efficiency in spreading information is the central theme of network scienc…
Hunan University（湖南大学）；National University of Singapore（新加坡国立大学）；Singapore Management University（新加坡管理大学）；Alibaba-Zhejiang University Joint Research Institute of Frontier Technologies（阿里巴巴-浙江大学前沿技术联合研究院）
A great variety of complex systems ranging from user interactions in communication networks to transactions in financial markets can be modeled as temporal graphs, which consist of a set of vertices and a series of timestamped and directed edges. Temporal motifs in temporal graphs are generalized from subgraph patterns in static graphs which take into account edge orderings and durations in addition to structures. Counting the number of occurrences of temporal motifs is a fundamental problem for temporal network analysis. However, existing methods either cannot support temporal motifs or suffe…
With the growing amount of available temporal real-world network data, an important question is how to efficiently study these data. One can simply model a temporal network as either a single aggregate static network, or as a series of time-specific snapshots, each of which is an aggregate static network over the corresponding time window. The advantage of modeling the temporal data in these two ways is that one can use existing well established methods for static network analysis to study the resulting aggregate network(s). Here, we develop a novel approach for studying temporal network data …
We develop random graph models where graphs are generated by connecting not only pairs of vertices by edges but also larger subsets of vertices by copies of small atomic subgraphs of arbitrary topology. This allows the for the generation of graphs with extensive numbers of triangles and other network motifs commonly observed in many real world networks. More specifically we focus on maximum entropy ensembles under constraints placed on the counts and distributions of atomic subgraphs and derive general expressions for the entropy of such models. We also present a procedure for combining distri…
Since network motifs are an important property of networks and some networks have the behaviors of rewiring or reducing or adding edges between old vertices before new vertices entering the networks, we construct our non-randomized model N(t) and randomized model N'(t) that have the predicated fixed subgraphs like motifs and satisfy both properties of growth and preferential attachment by means of the recursive algorithm from the lower levels of the so-called bound growing network models. To show the scale-free property of the randomized model N'(t), we design a new method, called edge-cumulat…
Zhejiang University of Technology（浙江工业大学）；NetEase Inc.（网易公司）；Zhejiang University（浙江大学）；City University of Hong Kong（香港城市大学）
Real-world networks exhibit prominent hierarchical and modular structures, with various subgraphs as building blocks. Most existing studies simply consider distinct subgraphs as motifs and use only their numbers to characterize the underlying network. Although such statistics can be used to describe a network model, or even to design some network algorithms, the role of subgraphs in such applications can be further explored so as to improve the results. In this paper, the concept of subgraph network (SGN) is introduced and then applied to network models, with algorithms designed for constructi…
The design of robust supply and distribution systems is one of the fundamental challenges at the interface of network science and logistics. Given the multitude of performance criteria, real-world constraints, and external influences acting upon such a system, even formulating an appropriate research question to address this topic is non-trivial. Here we present an abstraction of a supply and distribution system leading to a minimal model, which only retains stylized facts of the systemic function and, in this way, allows us to investigate the generic properties of robust supply networks. On t…
Querying Cohesive Subgraph regarding Span-Constrained Triangles on Temporal Graphs with Dynamic Index Maintenance
The Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州））；School of Computer Science, Wuhan University（武汉大学计算机学院）；The Hong Kong University of Science and Technology（香港科技大学）
