A connectedness constraint for learning sparse graphs

Abstract

Abstract-Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a graph highly sparse can split the graph into several disconnected components, leading to several separate networks. The main difficulty is that connectedness is often treated as a combinatorial property, making it hard to enforce in e.g. convex optimization problems. In this article, we show how connectedness of undirected graphs can be formulated as an analytical property and can be enforced as a convex constraint. We especially show how the constraint relates to the distributed consensus problem and graph Laplacian learning. Using simulated and real data, we perform experiments to learn sparse and connected graphs from data.

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“…Another example is the approach in [83] which learns a sparse graph with connected components. Learning graphs with desirable properties inspired by a specific application domain (e.g., community detection [2]) can also have great potential benefit, and it is a topic worth investigating.…”

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“…0 < c min < |L i, j |< c max gives the range of nonzero values of the off-diagonal entries of L. Constraint (10) encodes the property of a graph Laplacian described in Equation (2). It is known [11, Proposition 1] that any undirected graph is connected if and only if L + 1 n 1 T 1 is a positive-definite matrix, therefore Constraint (11) ensures that the graph represented by L must be connected. Constraint (12) and Constraint (13) models Π as the closed adjacency matrix of a graph.…”

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“…For improved interpretability and precise identification of the structure in the data, it is desirable to learn a graph with a specific structure. For example, gene pathways analysis are studied through multi-component graph structures [6,7], as genes can be grouped into pathways, and connections within a pathway might be more likely than connections between pathway, forming a cluster; a bipartite graph structure yields a more precise model for drug matching and topic modeling in document analysis [8,9]; a regular graph structure is suited for designing communication efficient deep learning architectures [10,11]; and a sparse yet connected graph structure is required for graph signal processing applications [12].…”