273,453 research outputs found
Higher order clustering coefficients in Barabasi-Albert networks
Higher order clustering coefficients are introduced for random
networks. The coefficients express probabilities that the shortest distance
between any two nearest neighbours of a certain vertex equals , when one
neglects all paths crossing the node . Using we found that in the
Barab\'{a}si-Albert (BA) model the average shortest path length in a node's
neighbourhood is smaller than the equivalent quantity of the whole network and
the remainder depends only on the network parameter . Our results show that
small values of the standard clustering coefficient in large BA networks are
due to random character of the nearest neighbourhood of vertices in such
networks.Comment: 10 pages, 4 figure
Tensor Spectral Clustering for Partitioning Higher-order Network Structures
Spectral graph theory-based methods represent an important class of tools for
studying the structure of networks. Spectral methods are based on a first-order
Markov chain derived from a random walk on the graph and thus they cannot take
advantage of important higher-order network substructures such as triangles,
cycles, and feed-forward loops. Here we propose a Tensor Spectral Clustering
(TSC) algorithm that allows for modeling higher-order network structures in a
graph partitioning framework. Our TSC algorithm allows the user to specify
which higher-order network structures (cycles, feed-forward loops, etc.) should
be preserved by the network clustering. Higher-order network structures of
interest are represented using a tensor, which we then partition by developing
a multilinear spectral method. Our framework can be applied to discovering
layered flows in networks as well as graph anomaly detection, which we
illustrate on synthetic networks. In directed networks, a higher-order
structure of particular interest is the directed 3-cycle, which captures
feedback loops in networks. We demonstrate that our TSC algorithm produces
large partitions that cut fewer directed 3-cycles than standard spectral
clustering algorithms.Comment: SDM 201
CHIEF : clustering With higher-order motifs in big networks
Clustering network vertices is an enabler of various applications such as social computing and Internet of Things. However, challenges arise for clustering when networks increase in scale. This paper proposes CHIEF (Clustering with HIgher-ordEr motiFs), a solution which consists of two motif clustering techniques: standard acceleration CHIEF-ST and approximate acceleration CHIEF-AP. Both algorithms firstly find the maximal -edge-connected subgraphs within the target networks to lower the network scale by optimizing the network structure with maximal -edge-connected subgraphs, and then use heterogeneous four-node motifs clustering in higher-order dense networks. For CHIEF-ST, we illustrate that all target motifs will be kept after this procedure when the minimum node degree of the target motif is equal or greater than . For CHIEF-AP, we prove that the eigenvalues of the adjacency matrix and the Laplacian matrix are relatively stable after this step. CHIEF offers an improved efficiency of motif clustering for big networks, and it verifies higher-order motif significance. Experiments on real and synthetic networks demonstrate that the proposed solutions outperform baseline approaches in large network analysis, and higher-order motifs outperform traditional triangle motifs in clustering. © 2022 IEEE Computer Society. All rights reserved
CHIEF: Clustering with Higher-order Motifs in Big Networks
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 optimizing the
network structure with maximal k-edge-connected subgraphs. For CHIEF-ST, we
illustrate that all target motifs will be kept after this procedure when the
minimum node degree of the target motif is equal or greater than k. For
CHIEF-AP, we prove that the eigenvalues of the adjacency matrix and the
Laplacian matrix are relatively stable after this step. That is, CHIEF-ST has
no influence on motif clustering, whereas CHIEF-AP introduces limited yet
acceptable impact. In the second procedure, we employ higher-order motifs,
i.e., heterogeneous four-node motifs clustering in higher-order dense networks.
The contributions of CHIEF are two-fold: (1) improved efficiency of motif
clustering for big networks; (2) verification of higher-order motif
significance. The proposed solutions are found to outperform baseline
approaches according to experiments on real and synthetic networks, which
demonstrates CHIEF's strength in large network analysis. Meanwhile,
higher-order motifs are proved to perform better than traditional triangle
motifs in clustering
Motif Clustering and Overlapping Clustering for Social Network Analysis
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 within the same
cluster, while "less relevant" social subgraphs are allowed to lie across
clusters. We then proceed to introduce the notion of motif covers, in which the
goal is to cover the vertices of motifs via the smallest number of (near)
cliques in the graph. Motif cover algorithms provide a natural solution for
overlapping clustering and they also play an important role in latent feature
inference of networks. For both motif correlation clustering and its extension
introduced via the covering problem, we provide hardness results, algorithmic
solutions and community detection results for two well-studied social networks
Beyond clustering: mean-field dynamics on networks with arbitrary subgraph composition
Clustering is the propensity of nodes that share a common neighbour to be connected. It is ubiquitous in many networks but poses many modelling challenges. Clustering typically manifests itself by a higher than expected frequency of triangles, and this has led to the principle of constructing networks from such building blocks. This approach has been generalised to networks being constructed from a set of more exotic subgraphs. As long as these are fully connected, it is then possible to derive mean-field models that approximate epidemic dynamics well. However, there are virtually no results for non-fully connected subgraphs. In this paper, we provide a general and automated approach to deriving a set of ordinary differential equations, or mean-field model, that describes, to a high degree of accuracy, the expected values of system-level quantities, such as the prevalence of infection. Our approach offers a previously unattainable degree of control over the arrangement of subgraphs and network characteristics such as classical node degree, variance and clustering. The combination of these features makes it possible to generate families of networks with different subgraph compositions while keeping classical network metrics constant. Using our approach, we show that higher-order structure realised either through the introduction of loops of different sizes or by generating networks based on different subgraphs but with identical degree distribution and clustering, leads to non-negligible differences in epidemic dynamics
Clustering Phase Transitions and Hysteresis: Pitfalls in Constructing Network Ensembles
Ensembles of networks are used as null models in many applications. However,
simple null models often show much less clustering than their real-world
counterparts. In this paper, we study a model where clustering is enhanced by
means of a fugacity term as in the Strauss (or "triangle") model, but where the
degree sequence is strictly preserved -- thus maintaining the quenched
heterogeneity of nodes found in the original degree sequence. Similar models
had been proposed previously in [R. Milo et al., Science 298, 824 (2002)]. We
find that our model exhibits phase transitions as the fugacity is changed. For
regular graphs (identical degrees for all nodes) with degree k > 2 we find a
single first order transition. For all non-regular networks that we studied
(including Erdos - Renyi and scale-free networks) we find multiple jumps
resembling first order transitions, together with strong hysteresis. The latter
transitions are driven by the sudden emergence of "cluster cores": groups of
highly interconnected nodes with higher than average degrees. To study these
cluster cores visually, we introduce q-clique adjacency plots. We find that
these cluster cores constitute distinct communities which emerge spontaneously
from the triangle generating process. Finally, we point out that cluster cores
produce pitfalls when using the present (and similar) models as null models for
strongly clustered networks, due to the very strong hysteresis which
effectively leads to broken ergodicity on realistic time scales.Comment: 13 pages, 11 figure
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